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The purpose of this study is to acquaint the reader with one of the effective approaches to describing processes in adaptive networks, built in the framework of the well-known Kuramoto model. Methods. The solution to this problem is based on the analysis of the results of works devoted to the study of the dynamics of oscillatory networks with adaptive couplings. Main classes of models of dynamical couplings used in the description of adaptive networks are considered, and the dynamical and structural effects caused by the presence of the corresponding law of coupling adaptation are analyzed. Results. Principles of constructing models of adaptive networks based on the phase description developed by Kuramoto are presented. Materials presented in the review show that the Kuramoto system with dynamic couplings demonstrates a wide range of fundamentally new phenomena and modes. Considered networks include well-known models of dynamical couplings that implement various laws of adaptation of inter-element interactions depending on the states of the elements, in particular, on their relative phase difference. For each network model, a class of possible solutions is established, and general properties of collective dynamics are identified, due to the presence of adaptability of couplings. One of the features of such networks is the multistability of behavior, determined by the possibility of the formation in the network of many different cluster states, including chimera ones. It was found that the implemented coupling adaptation mechanism affects not only the configuration of clusters formed in the network, but also the nature of phase distributions within them. The processes of cluster formation are accompanied by a restructuring of the interaction topology, leading to the formation of hierarchical and modular structures. Conclusion. In conclusion, we briefly summarize the results presented in the review.

We propose a robust universal approach to identify multiple network dynamical states, including stationary and travelling chimera states based on an adaptive coherence measure. Our approach allows automatic disambiguation of synchronized clusters, travelling waves, chimera states, and asynchronous regimes. In addition, our method can determine the number of clusters in the case of cluster synchronization. We further couple our approach with a new speed calculation method for travelling chimeras. We validate our approach by an example of a ring network of type II Morris-Lecar neurons with asymmetrical nonlocal inhibitory connections where we identify a rich repertoire of coherent and wave states. We propose that the method is robust for the networks of phase oscillators and extends to a general class of relaxation oscillator networks.

We considered a network consisting of two populations of phase oscillators, the interaction of which is determined by different rules for the coupling adaptation. The introduction of various adaptation rules leads to the suppression of splay states and the emergence of each population complex non-stationary behavior called transient circulant clusters. In such states, each population contains a pair of anti-phase clusters whose size and composition slowly change over time as a result of successive transitions of oscillators between clusters. We show that an increase in the mismatch of the adaptation rules makes it possible to stop the process of rearrangement of clusters in one or both populations of the network. Transitions to such modes are always preceded by the appearance of solitary states in one of the populations.

We study the influence of the initial topology of connections on the organization of synchronous behavior in networks of phase oscillators with adaptive couplings. We found that networks with a random sparse structure of connections predominantly demonstrate the scenario as a result of which chimera states are formed. The formation of chimera states retains the features of the hierarchical organization observed in networks with global connections [D.V. Kasatkin, S. Yanchuk, E. Scholl, V.I. Nekorkin, Phys. Rev. E 96, 062211 (2017)], and also demonstrates a number of new properties due to the presence of a random structure of network topology. In this case, the formation of coherent groups takes a much longer time interval, and the sets of elements that form these groups can be significantly rearranged during the evolution of the network. We also found chimera states, in which along with the coherent and incoherent groups, there are subsets, whose different elements can be synchronized with each other for sufficiently long periods of time.

Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system. We extend recent results on the appearance of hierarchical frequency-multi-clusters by investigating the effect of the time-scale separation. We show that the slow adaptation in comparison with the fast phase dynamics is necessary for the emergence of the multi-clusters and their stability. Additionally, we study the role of double antipodal clusters, which appear to be unstable for all considered parameter values. We show that such states can be observed for a relatively long time, i.e., they are metastable. A geometrical explanation for such an effect is based on the emergence of a heteroclinic orbit., 16 pages, 8 figures

We study the dynamical regimes in the system of two identical interacting phase oscillators with plastic couplings. The joint evolution of the states of the elements and the interelement couplings is a feature of the system studied. It is shown that the introduction of plastic couplings leads to a multistable behavior of the system and emergence of the asynchronous regimes which are not observed for the considered parameter values in the case of static couplings. The parameter plane is divided into regions with different dynamic regimes of the system. In particular, the regions in which the system demonstrates bistable synchronous behavior and the region in which the coexistence of many various asynchronous regimes is observed are singled out.

In a network of pulse-coupled oscillators with adaptive coupling, we discover a dynamical regime which we call an "itinerant chimera." Similarly as in classical chimera states, the network splits into two domains, the coherent and the incoherent. The drastic difference is that the composition of the domains is volatile, i.e., the oscillators demonstrate spontaneous switching between the domains. This process can be seen as traveling of the oscillators from one domain to another or as traveling of the chimera core across the network. We explore the basic features of the itinerant chimeras, such as the mean and the variance of the core size, and the oscillators lifetime within the core. We also study the scaling behavior of the system and show that the observed regime is not a finite-size effect but a key feature of the collective dynamics which persists even in large networks.

We report the phenomenon of self-organized emergence of hierarchical multilayered structures and chimera states in dynamical networks with adaptive couplings. This process is characterized by a sequential formation of subnetworks (layers) of densely coupled elements, the size of which is ordered in a hierarchical way, and which are weakly coupled between each other. We show that the hierarchical structure causes the decoupling of the subnetworks. Each layer can exhibit either a two-cluster state, a periodic traveling wave, or an incoherent state, and these states can coexist on different scales of subnetwork sizes.

We present a new approach for the description and analysis of sequential dynamics of excitatory neural networks. It is based on the reduction of dynamics of networks to the dynamics of cellular automata (CA) on the graph of connections. The attractors of the CA determine the different regimes of sequential dynamics of the original neural network. We illustrate our approach through Morris–Lecar and Hodgkin–Huxley neural networks.

The phenomenon of self-referential phase reset is investigated in ensembles of interacting FitzHugh–Nagumo neurons with different topologies of couplings. It is shown that the reset phase of neurons oscillation is independent of the initial phase and is determined by parameters of the stimulus. This process does not require direct influence on all elements of the ensemble and takes place when the stimulus is applied to one of the interacting neurons only. The influence of inter-neuron couplings and stimulus parameters on ensemble dynamics and phase reset phenomenon is studied.

We propose a new model of synaptic transmission on the basis of a FitzHugh–Nagumo system with nonlinear recovery. It is shown that the Morrice–Lecar neuron ensemble formed by such couplings shows various structurally stable regimes of transient dynamics in the form of sequential transitions between various metastable oscillatory states of the system.

The processes of synchronization and phase cluster formation are investigated in a complex network of dynamically coupled phase oscillators. Coupling weights evolve dynamically depending on the phase relations between the oscillators. It is shown that the network exhibits several types of behavior: the globally synchronized state, two-cluster and multi-cluster states, different synchronous states with a fixed phase relationship between the oscillators and chaotic desynchronized state.

We discuss the problems of generation of chaotically modulated oscillations in small ensembles of coupled self-excited oscillators with phase control. Special attention is paid to analyzing the regions of generation of chaotic oscillations in parameter space. It is shown that transition to collective dynamics allows us to efficiently solve the problem of generation of chaotically modulated oscillations in a sufficiently wide parameter-space region.

The generation of chaotically modulated oscillations in autooscillatory systems with frequency control is considered. It is shown that, by coupling two such systems, it is possible to considerably expand a region corresponding to the generation of chaotic oscillations in the space of parameters of this ensemble and to provide additional means of controlling the characteristics of oscillations by changing the parameters of coupling.

In this paper, we consider the dynamics of two types of modular neural networks. The first network consists of two modules of non-interacting neurons while each neuron inhibits all the neurons of an opposite module. We explain the mechanism for emergence of anti-phase group bursts in the network and showed that the collective behavior underlies a regular response of the system to external pulse stimulation. The networks of the second type contain modules with complex topology which are connected by relatively sparse excitatory delayed coupling. We found a dual role of the inter-module coupling delay in the collective network dynamics. First, with increasing time delay, in-phase and anti-phase regimes, where individual spikes form rhythmic modular burst-like oscillations, alternate with each other. Second, the average frequency of the collective oscillations in each of these regimes decreases with increasing inter-module coupling delay.

We propose a model of neural network demonstrating variety of sequential activity modes. Unlike the previously known models of transient dynamics, in the present model transient sequential modes are formed by means of dynamical bifurcations and not directly related to the existence of heteroclinic channels. It is shown that network being initially at rest generates a sequence of metastable oscillatory states of activity in response to an external stimulus. We study the influence of the parameters characterized the initial times of synaptic activation processes caused by input information signals on the network dynamics. Numerical simulation of the model has shown, that these parameters determine not only the structure of the set of oscillatory metastable states and the sequence of transitions between them, but also the temporal characteristics of the transition sequences such as the time duration of the oscillatory metastable states.

We have developed a new approach for the description of sequential dynamics of excitatory neural networks. Our approach is based on the dynamics of synapses possessing the short-term plasticity property. We suggest a model of such synapses in the form of a second-order system of nonlinear ODEs. In the framework of the model two types of responses are realized-the fast and the slow ones. Under some relations between their timescales a cellular automaton (CA) on the graph of connections is constructed. Such a CA has only a finite number of attractors and all of them are periodic orbits. The attractors of the CA determine the regimes of sequential dynamics of the original neural network, i.e., itineraries along the network and the times of successive firing of neurons in the form of bunches of spikes. We illustrate our approach on the example of a Morris-Lecar neural network.